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In
mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ...
, Lambert summation is a summability method for a class of
divergent series In mathematics, a divergent series is an infinite series that is not convergent, meaning that the infinite sequence of the partial sums of the series does not have a finite limit. If a series converges, the individual terms of the series must ...
.


Definition

A series \sum a_n is ''Lambert summable'' to ''A'', written \sum a_n = A \,(\mathrm), if :\lim_ (1-r) \sum_^\infty \frac = A . If a series is convergent to ''A'' then it is Lambert summable to ''A'' (an
Abelian theorem In mathematics, Abelian and Tauberian theorems are theorems giving conditions for two methods of summing divergent series to give the same result, named after Niels Henrik Abel and Alfred Tauber. The original examples are Abel's theorem showing that ...
).


Examples

* \sum_^\infty \frac = 0 \,(\mathrm), where μ is the
Möbius function The Möbius function is a multiplicative function in number theory introduced by the German mathematician August Ferdinand Möbius (also transliterated ''Moebius'') in 1832. It is ubiquitous in elementary and analytic number theory and most oft ...
. Hence if this series converges at all, it converges to zero.


See also

*
Lambert series In mathematics, a Lambert series, named for Johann Heinrich Lambert, is a series taking the form :S(q)=\sum_^\infty a_n \frac . It can be resumed formally by expanding the denominator: :S(q)=\sum_^\infty a_n \sum_^\infty q^ = \sum_^\infty b_m ...
*
Abel–Plana formula In mathematics, the Abel–Plana formula is a summation formula discovered independently by and . It states that :\sum_^\infty f(n)=\frac 1 2 f(0)+ \int_0^\infty f(x) \, dx+ i \int_0^\infty \frac \, dt. It holds for functions ''f'' that are holo ...
*
Abelian and tauberian theorems In mathematics, Abelian and Tauberian theorems are theorems giving conditions for two methods of summing divergent series to give the same result, named after Niels Henrik Abel and Alfred Tauber. The original examples are Abel's theorem showing that ...


References

* * * Mathematical series Summability methods {{Mathanalysis-stub